Rational curves and Seshadri constants on Enriques surfaces (Q6566369)

General Enriques surfaces do not contain smooth rational curves but, since they admit many elliptic pencils, they contain nodal rational curves, which in fact are numerically \(2\)-divisible. It has been recently known (see the Introduction of the paper under review and references therein), as a consequence of a similar result on \(K3\)-surfaces, that Enriques surfaces contain irreducible rational curves whose self-intersection tends to infinity. The first main result of the paper under review states that, for any very general Enriques surface \(S\), all irreducible rational curves on \(S\) are numerically \(2\)-divisible. The main ingredient of the proof is to degenerate the surface to a union of an elliptic ruled surface and a blown-up plane to exclude the posibility of the rational curves not to be \(2\)-divisible. This result is applied in Thm. 1.3 to prove, for \(H\) a nef and big divisor on \(S\), an interesting equality between its Sehsadri constant \(\epsilon(H)\) and the so called \(\phi(H)\), defined as the minimum of \(H\cdot E\) for \(E>0\), \(E^2=0\). The proof relies on the known fact that the Seshadri constant is computed, in Enriques surfaces, via an irreducible curve and on the fact that irreducible curves \(C\) providing quotients (of their degree \(C\cdot H\) and the multiplicity of some of its points) smaller than \(\phi(H)\) have to be rational. Hence, in the possible list of such curves (which is finite), only the \(2\)-divisible case (which is unique) can appear, as a consequence of Thm. 1.1. Finally, this case can be excluded using the same degeneration trick.